What triggers a 'TWEAK' event?
I was able to get 'EVT_TWEAK_L' to print the message from within the operator by changing the keymap. 

I changed the way the operator was called in the keymap from: 


  Mouse> Any Left Mouse


to


  Tweak> Left> Any


I assumed that since a modal operator called by a LMB click could also catch a bunch of other events such as 'RIGHTMOUSE', 'MOUSEMOVE', 'ESC', alt, ctrl, etc etc, that it would also be able to catch tweak events, but that doesn't seem to be the case.

I'll update the answer if I find another way to do it.